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Permutations with Repetition Calculator

2026-09-29 8 min

Ordered draws with replacement: n^r outcomes.

Grade level: 8–12; Curriculum band
Difficulty: Intermediate; For this topic
Time to master: 2 weeks; With daily practice
Tools on page: 10 on page; Calculator + practice

Counts — Permutations with Repetition Calculator

Options (n)

Draws (r)

Results update live as you type

Status:Count 125

Result Summary

Count125

Results

Optimal

Count

125

Exact integer count

Multiplicative formulas avoid factorial overflow; integer guards enforce ranges.

Statistics

Live readouts from your current counts.

Count

125

exact outcomes

Visualizations

Magnitude (log10)

2.1log10

Digits

digits3

Formula

Ordered draws with replacement: n^r outcomes.

Outcomes = n^r

Alternative Form

Replacement keeps n options each draw

Variables

Symbol	Name	Unit	Description
n	Pool size	count	Total items
k	Selection	count	Chosen or arranged
result	Count	outcomes	Exact integer

Explanation

Order decides P vs C; replacement decides powers; partitions divide by symmetries.

Derivation

Factorial products count sequences; dividing by k! forgets order; stars-and-bars encodes multisets; recurrences build Stirling, Catalan and partition numbers.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Validate ranges and integer conditions.

Result: Inputs valid

Step 2

Select the counting model for this question.

Outcomes = n^r

Result: Model chosen

Step 3

Evaluate products term by term.

Result: Result = 125

Step 4

Cross-check with the companion identity.

Replacement keeps n options each draw

Result: Check passed

Step 5

Round to exact integer for display.

Result: Display ready

Step 6

State the finished count.

Result: Result = 125

Final Answer

125

Worked Examples

See how the calculator works with different scenarios.

Scenario

Run the page defaults once.

Inputs

Inputs:defaults

Calculation

Apply the formula directly

Result

See live result

Interpretation

Confirms the engine path end to end.

Understanding Your Result

Your Result: Count: 125

Typical range: C(10,3) = 120; 6! = 720; Catalan(5) = 42.

What This Means

The headline integer is the exact number of outcomes under the stated rules.

Why It Matters

Passwords, lotteries, schedules and samples all price possibilities with these counts.

Professional Interpretation

Name order/replacement assumptions whenever you quote a count.

Recommendations

•Decide order first
•Decide replacement second
•Use logs for giant counts

Learn — The Complete Guide

Permutations with Repetition Calculator counts without listing — formulas replace exhaustive enumeration.

The core concept

Multiplication for stages; division for forgotten order; recurrences for structured families.

Order decides P vs C
Replacement decides powers
Divide out symmetries

Derivations — why the formulas work

Why divide by k!

1P counts orders
2Each set has k! orders
3Divide: C = P/k!

Where you meet this in real life

▸Lotteries and cards

▸Passwords

▸Scheduling

▸Sampling plans

▸Lotteries

▸Passwords

▸Scheduling

▸Quality sampling

Learn this first — prerequisites in order

1Factorials
2Fractions
3Logic cases

Careers that use this daily

•Analyst — sampling
•Security — passwords
•Coach — lineups
•Planner — schedules

Why it helps

•Overflow-safe products
•Symmetry cross-checks
•Live magnitude view

Best practices

•Classify P/C first
•Check k <= n
•Verify with small cases

Limitations — what this tool does not do

•Exact integers only in range
•Giant counts need logs

Curriculum map — where this topic lives in school

Level	Grades	Standards	Skills
Middle school	7–8	7.SP	Counting principle · Factorials
High school	9–12	HSS-CP	P and C · Binomial links

Timeline — how this idea grew up

1654
Pascal triangle
Binomial counts systematized.
1830s
Catalan numbers
Parentheses problem posed.

A little history

• Euler partitioned integers.
• Bell and Stirling classified set divisions.

Exam-day checklist

• Classify first
• Check bounds
• Verify tiny cases

Myths vs facts

Myth: Order never matters.

Truth: Rankings and codes are permutations, not combinations.

Myth: Bigger n means slightly bigger counts.

Truth: Factorial growth explodes past intuition.

Practical Tips

•Ask order first
•Ask replacement second
•Try small n by hand
•Use symmetry to halve work
•Copy with share bar

Common Mistakes

P vs C mixup

Rankings use P; committees use C.

Forgetting replacement

Dice/PINs repeat: use powers.

Overflow panic

Read the log10 magnitude bar.

Assumptions

This calculator makes the following assumptions to provide estimates:

Distinct items: Pool members distinguishable unless grouped.

Fair counting: Each outcome equally possible.

Exact ranges: Inputs inside stated bounds.

Limitations

Be aware of these limitations when using this calculator:

Range caps: Guards prevent overflow/DoS.

Integers only: Fractional n/k blocked.

Frequently Asked Questions

Answers to common questions about this calculator.

When replacement?

Dice rolls, PIN digits, repeated spins.

Growth?

Exponential in r.

How do I use this calculator?

Enter n, k or lists as labeled; the count and steps update live.

Does it show steps?

Yes — six traced steps plus a breakdown table mirror every number on this page.

Can I print or share results?

Use the share bar to copy a link or results, download CSV/JSON, or print the worksheet.

Is my data stored?

No — calculations run locally; history stays in your browser only.

Related Calculators

Explore more calculators related to this topic.

Related Guides

Deepen your understanding with these educational resources.

P vs C in 60 seconds

Order decides everything.

Checking counts

Small cases and symmetry.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Permutation: Ordered selection.
Combination: Unordered selection.
Factorial: Product 1..n.
Stars and bars: Multiset encoding.
Recurrence: Count from smaller counts.

Authoritative References

Trusted sources and standards referenced by this calculator.

Wikipedia — Combinatorics Math Is Fun — Combinatorics Khan Academy — Counting Brilliant — Counting

Applicable Standards

Common Core HSS-CP: counting and probability

Diagrams & Illustrations

Visual intuition for every core idea. Select a card to enlarge it.

Did You Know?

Short intuition builders — great for class warm-ups.

52! is cosmic

Card shuffles exceed atoms: 8 x 10^67.

Derangements hover at 1/e

About 37% of random permutations fix nothing.

Catalan hides everywhere

Parentheses, trees, paths — same numbers.

Compare Side by Side

Exact count plus scale views for giant numbers.

View	Measure	Value	Note
Count	125	exact	this page
log10	2.1	scale	digits - 1
Digits	3	length	decimal digits

Recent Calculations

Your last runs on this page — stored only in this browser, never uploaded.

Permutations with Repetition Calculator: 125
Current inputs — live

Study Plan

A week-by-week path from prerequisites to mastery. Your checklist progress saves in this browser.

Prerequisites

▸Factorial notation
▸Fractions
▸Order vs chaos

Mastery progress — 0%

I pick P vs C instantly
I handle repetition
I check k <= n

1
Week 1 · ~15 min/day
P vs C fluency
Goals
- Classify order correctly
- Handle replacement
- Verify with small cases
Practice
Ten classifications daily.

Exam tips

•Write the model first
•Test tiny n

Practice Problems

0/3 solved

Try each problem first, peek at the hint only if stuck, then reveal the solution and mark it solved.

Ceasy

C(10, 3)?

Peasy

P(10, 3)?

Dmedium

!4?

Printable Worksheet

Generate a fresh problem set for class or homework. Print it, or copy the answer key separately.

Problems Name

Arithmetic practice — set #1Name: __________ · Date: __________

1.18 × 47 =
2.8 ÷ 11 =
3.38 × 26 =
4.35 ÷ 4 =
5.21 − 16 =
6.2 + 26 =
7.14 ÷ 12 =
8.23 ÷ 3 =
9.49 ÷ 7 =
10.22 − 17 =
11.41 + 18 =
12.47 − 35 =

Formula Cheat Sheet

Every formula on this page in one printable table.

Formula	Meaning	Example
C(n,k) = n!/(k!(n-k)!)	Choose	C(10,3) = 120
P(n,k) = n!/(n-k)!	Arrange	P(10,3) = 720
!n ~ n!/e	Derange	!5 = 44

Tip: every interactive block on this page — checklist, practice, worksheet, history — works offline and saves progress in your browser.

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Page Information

Last Updated:

2026-09-29

Version:

3.1.0-math

References:

License:

Free to use for educational and professional purposes.